Reconstructing a $f(R)$ theory from the $\alpha$-Attractors

We show an analogy at high curvature between a $f(R) = R + aR^{n - 1} + bR^2$ theory and the $\alpha$-Attractors. We calculate the expressions of the parameters $a$, $b$ and $n$ as functions of $\alpha$ and the predictions of the model $f(R) = R + aR^{n - 1} + bR^2$ on the scalar spectral index $n_{\rm s}$ and the tensor-to-scalar ratio $r$. We find that the power law correction $R^{n - 1}$ allows for a production of gravitational waves enhanced with respect to the one in the Starobinsky model, while maintaining a viable prediction on $n_{\rm s}$. We numerically reconstruct the full $\alpha$-Attractors class of models testing the goodness of our high-energy approximation $f(R) = R + aR^{n - 1} + bR^2$. Moreover, we also investigate the case of a single power law $f(R) = \gamma R^{2 - \delta}$ theory, with $\gamma$ and $\delta$ free parameters. We calculate analytically the predictions of this model on the scalar spectral index $n_{\rm s}$ and the tensor-to-scalar ratio $r$ and the values of $\delta$ which are allowed from the current observational results. We find that $-0.015 < \delta < 0.016$, confirming once again the excellent agreement between the Starobinsky model and observation.